Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $q = \dfrac{10a - 15}{a} \div \dfrac{10(2a - 3)}{-2} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{10a - 15}{a} \times \dfrac{-2}{10(2a - 3)} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (10a - 15) \times -2 } { a \times 10(2a - 3) } $ $ q = \dfrac {-2 \times 5(2a - 3)} {a \times 10(2a - 3)} $ $ q = \dfrac{-10(2a - 3)}{10a(2a - 3)} $ We can cancel the $2a - 3$ so long as $2a - 3 \neq 0$ Therefore $a \neq \dfrac{3}{2}$ $q = \dfrac{-10 \cancel{(2a - 3})}{10a \cancel{(2a - 3)}} = -\dfrac{10}{10a} = -\dfrac{1}{a} $